Arc of a ball and the two lines approach

The exploration of the arc of a ball began with the students attempting to trace the path of a ball arcing through the air in front of the white board. 

The students were then asked to find an equation for the arc of the ball using Desmos. Earlier in the week the students had used a quadratic equation to obtain the acceleration of gravity. The students also have had experience with linear and square root relationships. I ask the students to work in groups and give them fifteen to twenty minutes to come up with an equation.

One group was working with a direct linear relationship. All of the laboratories to date in physical science with a linear relationship had a y-intercept of zero. The students were not used to thinking in terms of a non-zero y-intercept. 

A y-intercept, however, does not help in this situation, 

This term no student realized that the path forward would require a quadratic equation, a curve, to get through all three points. All three groups were working with some form of linear equation.

With all three groups stalled and no longer able to make any progress I demonstrated that a line with a y-intercept would generate a line that passed through two of the three points including the vertex of the arc. 

There is a second line that also goes through the y-intercept and the other x-intercept.

If the equations for the two lines are multiplied together, the result is a curve that passes through both x-intercepts and which crosses the y-axis. The only issue is that the line crosses the y-axis up at 2500 instead of the desired y=50 crossing point.  

Noting that getting from 2500 down to 50 can be accomplished by multiplying by 50 ÷ 2500 or 0.02, one can rescale the graph by multiplying by 0.02. The result is the equation of the arc of the ball, passing through (-35, 0), (0, 50), and (35,0).  The coefficient on the x term is the result of 50 ÷ 35. 

I then showed the class that WolframAlpha can expand this.

Which produces the simpler equation 50 - 2x² ÷ 49.

The simpler equation can be shown to also go through the three points of the arc.

Of course Desmos can also regress a quadratic against all three coordinates and directly obtain the coefficients. The lead coefficient of 0.0408163 is the result of 50 ÷ 35². 

While one could obviously construct the equation more quickly from the vertex form of the quadratic equation, explaining how to obtain the value of a would require solving the equation at one of the roots. 

My intention above was two-fold. One was to find a way from linear equations to the quadratic equation as purely graphically as possible - with as little use of algebraic algorithms as possible. Allowing Desmos to make the linear regressions enables this non-calculational approach. The second was to push the students to continue to see that one can move from shapes on a graph to equation - and there is not necessarily one single way to do this. 

Another approach is just to go with sliders all the way, but that may take more time and will likely only result in an approximation for the value of a. 

While the demonstration of going from two lines to a quadratic undoubtedly lost the students along the way, the course will continue to move from data and the shapes the data creates to equations in the laboratories ahead. Mathematics and algebra as currently taught focuses on manipulating abstract symbols on a page - factoring, distributing, solving, applying algorithms. Calculations and symbol manipulations are often done by hand. Yet none of this leads students to think mathematically. To think, "Oh, there are three non-collinear points, an equation with an x-squared is needed to go through the three points." With the arrival of algebra engines and now of AI the details of the specific manipulations can be left to technology. Humans will need to move up to a higher level of working with the concepts in mathematics. 

And what does AI say about the above?

Given the prompt:

Find the function that goes through (-35,0), (0,50), and (35,0)

OpenAI ChatGPT responded:

Solving three equations in three unknowns using three coordinates. Yet another approach. And while students who are non-STEM majors will never have to solve a quadratic equation in their working life, physical science at least gets to illustrate the systems that obey a quadratic equation. Such as the arc of a ball.